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Did you read the proper class section before posting the thread?? Because a proper class can contain all large cardinals without conflicting with itself, where V is a hierarchical class in the model and class has delegate is L.Bro, the extreme height here is Inaccessible cardinal, why is that? Because this is just the context of "Transfinite progression to Large Cardinal" which means that the extreme height here is Inaccessible itself.
If you really want to upgrade to the top of the large cardinal or cover all the large cardinals, provide supportive follow-up volume.
You have to clarify why the context of large cardinal is not clear, since NBG set theory is said to have V and class correctly applied.